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On the Hasse principle for cubic surfaces. (English) Zbl 0151.03405
E. S. Selmer has conjectured [Math. Scand. 1, 113–119 (1953; Zbl 0051.03202)] that every diagonal cubic form in four variables which represents 0 in every \(p\)-adic field represents 0 in the rational field \(\mathbb Q\). The authors disprove the conjecture by the example \(5x^3+12y^3+9z^3+10t^3\). The proof that this form does not represent 0 in \(\mathbb Q\) uses properties of the group of ideal classes of the field \(\mathbb Q\left(\root 3\of {30}, \root 3\of {90}\right)\). The necessary information about the class number of \(\mathbb Q\left(\root 3\of a, \root 3\of b\right)\) is conveniently gathered in an appendix.

MSC:
11D25 Cubic and quartic Diophantine equations
14G05 Rational points
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
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